‘The Sapphires” is an unpolished gem of a musical, a dramedy with a familiar ’60s girl-group-on-the-rise story pasted over a backdrop of Australian racism and America’s long war in Vietnam.

It’s a tribute to the filmmakers (director Wayne Blair, working from a Tony Briggs and Keith Thompson script) that this confection often manages to connect the jaunty, sassy musical elements to the serious comment on Australian history.

A prologue captures a group of Aboriginal girls singing for family and friends in the Outback of the 1950s. This was an era when Australia routinely “stole” light-skinned aboriginal children to be raised in institutions and taught “white ways.” So that quartet, when we next see them 10 years later, is only a trio.

Gail, given a sneering fierceness by Deborah Mailman of “Bran Nue Day” and “Rabbit-Proof Fence,” plays guitar and bosses sister Cynthia (Miranda Tapsell) around. They’re off to a town talent show, if only they can shake their underage songbird baby sister Julie (Jessica Mauboy of “Bran Nue Day").

The talent show, where Gail and Cynthia sing a killer Merle Haggard cover, is where we meet the environment they’re trying to rise above. “No Abos,” one white Aussie yells. “Bloody monkeys,” growls another. The only guy to notice their talent is the MC of the show, a drunken Irish piano player named Dave (Chris O’Dowd from “Bridesmaids”).

They’ve seen an ad for singers needed to entertain American GIs in Vietnam, and the sisters (Julie, too) want Dave’s help to get to the audition and win the job. But Gail’s only going to take so much advice from a white boy.

There’s a cute montage of them polishing the act, learning to sing soul music, learning to sing “blacker.” The missing fourth from their childhood quartet (Shari Sebbens) joins up.

And then it’s off to Vietnam, where “Heard It Through the Grapevine” and “I’ll Take You There” take them from seedy Saigon clubs to hospitals the front lines at a fire base in the middle of nowhere. Along the way, they face love, blood and death, and come to terms with their heritage.

There’s more than a hint of the ’90s Roddy Doyle adaptation “The Commitments” in all this — people far removed from Memphis and Detroit connecting to soul music on a spiritual level. You have to take the “inspired by a true story” label with a roll of the eyes, as the film’s history is sloppy — singing songs that hadn’t yet been written at the time the film is set, singing to military units that never were in Vietnam.

But the clashing personalities make it work and the singing makes it sing. O’Dowd makes a terrific comic foil to the quartet — who fall into and yet rise above the cliché “types” that each girl represents — the one with the talent, the one with the libido, the one with the chip on her shoulder, etc.